Patrick Speissegger
Let $M \subseteq \H^{>0}$ be a pure scale on standard power domains. In this post, I gave the base step of the construction of a qaa field $(\F,L,T)$ as claimed here. The goal of this post is to finish this construction. Step 0.5: apply a $\log$-shift to the qaa field $(\F_0,L_0,T_0)$, that is, set $$\F’_1…
Let $\H$ be the Hardy field of $\Ranexp$, and let $M$ be a multiplicative $\RR$-subvector space of $\H^{>0}$; I continue to assume in this post that $M$ is a pure scale. A germ $h \in \H^{>0}$ is small if $h(x) \to 0$ as $x \to +\infty$. The construction discussed here works for the following type…
In order to address some of the questions raised in this post, I introduce here some relevant definitions and recast the construction of Ilyashenko algebras based on $\log$ monomials using these new notions. As before, let $\H$ be the Hardy field of germs at $+\infty$ of all $h:\RR \into \RR$ definable in $\Ranexp$, and denote…
The first version of my preprint on quasianalytic Ilyashenko algebras is available on arXiv. Feel free to leave comments here! Here is the abstract: I construct a quasianalytic field $\F$ of germs at $+\infty$ of real functions with logarithmic generalized power series as asymptotic expansions, such that $\F$ is closed under differentiation and log-composition; in…
This post provides the details for the Divisibility lesson. Let $F$ be a field and $f,g \in F[X]$. Definition 1 $f$ divides $g$, written $f|g$, if $g = hf$ for some $h ∈ F[X]$. We also say that $g$ is a multiple of $f$. If $\deg(f) \ge 1$, we say that $f$ is prime or…